LemonmathSelf-study sheet

Grade 9 · Lesson 6.4

Solving word problems by forming equations

Name: Date: Time spent: min

Goals

Know the steps for solving a problem by forming an equation.
Express unknown quantities in terms of one unknown and form a quadratic equation from the problem.
Solve problems about geometry, rates of work, motion and percentage growth.
Check the roots against the condition and against reality before concluding.

Remember

Three steps: form the equation (unknown, condition, express, equation) → solve → check against the condition and reality (positive lengths, natural numbers, …), then conclude.
Geometry: use perimeter, area and Pythagoras' theorem; draw a figure with dimensions in terms of $x$ .
Rate of work and motion: $\text{time}=\dfrac{\text{total}}{\text{rate}}$ or $\dfrac{\text{distance}}{\text{speed}}$ ; make the units consistent.
Growth: after two years a quantity is multiplied by $(1+r)^2$ .

Exercises

Exercise 1

Find two consecutive even natural numbers whose product is $224$ .

Exercise 2

A right-angled triangle has hypotenuse $13$ cm, and its two shorter sides differ by $7$ cm. Find the lengths of the two shorter sides.

Exercise 3

According to plan, a production team must make $600$ products, the same number each day. Thanks to a technical improvement, the team makes $10$ more products each day and finishes $2$ days ahead of plan. How many products per day did the plan require?

Exercise 4

A motorboat travels $35$ km downstream and then $35$ km upstream, taking $6$ hours in total (not counting stops). The current flows at $2$ km/h. Find the speed of the boat in still water.

Exercise 5

Suppose the price of a mobile phone falls from $5$ million dong to $4.05$ million dong after two reductions, each by the same percentage of the price just before it. By what percentage is the price reduced each time?

Exercise 6

A rectangular photo measuring $12$ cm $\times$ $8$ cm is put in a frame with a border of constant width. The total area of the frame and photo is $140$ cm². Find the width of the border. (Revise finding roots mentally.)

Exercise 7

(2.0 marks)

a) Solve the equation $x^2-2x-35=0$ .

b) Solve the following problem by forming an equation: A group of students hires a coach for a trip for $4.2$ million dong, shared equally. On the day of the trip, $2$ more students join, so each student pays $10$ thousand dong less than expected. How many students were in the group at first?

c) How much does each student actually pay?

Exercise 8 Challenge

Suppose a cinema sells $300$ tickets a day at a price of $60$ thousand dong. A survey shows that for every $5$ thousand dong cut from the price, it sells $50$ more tickets a day. The cinema wants a daily revenue of $20$ million dong and chooses the option that sells more tickets. What is the new ticket price?

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. The product of two consecutive natural numbers is

110

. If the smaller number is

x

, which equation is correct?

x(x+1)=110

.B.

x+(x+1)=110

.C.

x^2+1=110

.D.

2x+1=110

Question 2. A car travels

90

km at a speed of

x

(km/h). If it went

15

km/h faster, it would arrive

1

hour earlier. Which equation is correct?

\dfrac{90}{x}-\dfrac{90}{x+15}=1

.B.

\dfrac{90}{x+15}-\dfrac{90}{x}=1

.C.

\dfrac{90}{x}+\dfrac{90}{x+15}=1

.D.

90x-90(x+15)=1

Question 3. In a problem where the unknown

x

is a width (m), we get the equation

x^2+3x-70=0

with roots

7

and

-10

. The width is

7

m.B.

-10

m.C.

7

m or

10

m.D.

10

Question 4. The price of a product rises from

200

thousand dong to

242

thousand dong after two increases, each by

r\%

of the previous price. The value of

r

10

.B.

21

.C.

10.5

.D.

11

Part II. True or false

Question 1. A rectangular garden is

4

m longer than it is wide and has area

96

m². Let the width be

x

(m),

x>0

. Decide whether each statement is true or false.

a) The length of the garden is $x+4$ (m).	T / F
b) We have the equation $x^2+4x-96=0$ .	T / F
c) The width of the garden is $12$ m.	T / F
d) The perimeter of the garden is $40$ m.	T / F

Part III. Short answer

Question 1. The product of two consecutive even natural numbers is

288

. Find the larger number.

Question 2. Suppose the population of a ward grows from

20\,000

20\,808

people in

2

years, at the same rate of

r\%

each year. Find

r

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: two consecutive even numbers differ by $2$ .

Steps:

Let the smaller number be $x$ and the larger $x+2$ .
Solve $x(x+2)=224$ .

Exercise 2.

What you need: Pythagoras' theorem.

Steps:

Let the shortest side be $x$ (cm), $x>0$ ; the other is $x+7$ .
Solve $x^2+(x+7)^2=169$ .

Exercise 3.

What you need: $\text{number of days}=\dfrac{\text{total number of products}}{\text{rate}}$ .

Steps:

Let the planned rate be $x$ (products per day), $x\in\mathbb{N}^*$ .
Form the equation $\dfrac{600}{x}-\dfrac{600}{x+10}=2$ .

Exercise 4.

What you need: speed downstream $=x+2$ , speed upstream $=x-2$ .

Steps:

Let the speed in still water be $x$ (km/h), $x>2$ .
Form the equation $\dfrac{35}{x+2}+\dfrac{35}{x-2}=6$ and solve it.

Exercise 5.

What you need: each reduction by $r$ multiplies the price by $1-r$ .

Steps:

Form the equation $5(1-r)^2=4.05$ with $0<r<1$ .
Solve it as $X^2=k$ .

Exercise 6.

What you need: the whole frame is a rectangle $(12+2x)\times(8+2x)$ ; find the roots mentally when $a+b+c=0$ .

Steps:

Let the width of the border be $x$ (cm), $x>0$ .
Form the equation, simplify it, then find the roots mentally.

Exercise 7.

What you need: the reduced discriminant; the amount per student $=\dfrac{\text{total cost}}{\text{number of students}}$ .

Steps:

a) Work out $\Delta'$ .
b) Work in thousands of dong: $4.2$ million $=4200$ thousand. Let the original number of students be $x$ and form $\dfrac{4200}{x}-\dfrac{4200}{x+2}=10$ .
c) Divide the total cost by $x+2$ .

Exercise 8.

What you need: revenue $=$ price $\times$ number of tickets; the number of tickets and the price depend on each other.

Steps:

Let the number of $5$ thousand dong cuts be $x$ ( $x\in\mathbb{N}$ , $x<12$ ).
The price is $60-5x$ and the number of tickets is $300+50x$ ; form the equation with revenue $20\,000$ thousand dong.
Choose the root that gives more tickets.

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